Geometric Interpretation of Tetraprimes

Following on from my previous post titled Tetraprimes, I asked Gemini the following:

I’d like you to create a program in SageMath that will accept any positive integer with up to four not necessarily distinct prime factors as INPUT and if the integer has FOUR prime factors it should return 4D Hypervolume, 3D Surface Volume, 2D Total Face Area and 1D Total Edge Length. If it has THREE factors then 3D Volume, 2D Surface Area and 1D Edge Length are returned. If it has TWO factors then 2D Surface Area and 1D Edge Length are returned. If the input is prime or has more than four factors then this is stated and no output is returned. The factorisation of the numbers shown in the output should also be displayed.

Here is a permalink to the program that it created using 28302 as input. The output looks of the program is shown below:

Input: 28302
4D Hypervolume: 28302 = 2 * 3 * 53 * 89
3D Surface Volume: 48874 = 2 * 7 * 3491
2D Total Face Area: 21732 = 2^2 * 3 * 1811
1D Total Edge Length: 1176 = 2^3 * 3 * 7^2

The program also deals with triprimes and biprimes. Here is the output using 28303 and 28299 as inputs.

Input: 28303
3D Volume: 28303 = 11 * 31 * 83
2D Surface Area: 7654 = 2 * 43 * 89
1D Edge Length: 500 = 2^2 * 5^3

Input: 28299
2D Surface Area: 28299 = 3 * 9433
1D Edge Length: 18872 = 2^3 * 7 * 337

I've incorporated this algorithm into my daily number analysis. 

In the example shown above, we can see easily that 28303 can be interpreted as the volume of a rectangular prism with a surface area that is also a sphenic number and thus interpretable as the volume of another rectangular prism. This additional information for my daily number analysis will prove useful in other ways I'm sure.